As far as anime and manga are concerned, a nosebleed indicates that a character is experiencing strong feelings of lust (usually for a particular person, but not necessarily). One theory behind this symbol is that the rapid acceleration of one's heartbeat causes a rise in blood pressure, which causes the afflicted one's nose to bleed. Another is that nosebleeds are a metaphor for ejaculation, as lj has pointed out below.
It should be noted that nosebleeds usually happen to male characters upon seeing certain female characters, particularly if said female character is exhibiting large amounts of fan service. I've yet to see a female character nosebleed, although I'm sure it has happened at some point. (This gives credence to the ejaculation theory, which is starting to sound more correct all the time. I salute you, lj.)
The nosebleed is one of a group of stylized indicators to denote emotion in anime and manga, and is usually used in more light-hearted moments, particularly with super deformed art. Other indicators include the vein-throb, the sweat-drop and the facefault.